0

I have a logit model that I am constructing, and I have seen different explanations on the interpretation of the beta coefficients. I think I understand both mathematically, but wanted clarification and insight as to which one to use. My Wooldridge textbook, and many other sources, express the logit function as $P(y=1\left|x_1...x_k\right)=G\left(z\right)$, where $z=β_o+β_1+...+β_k$, and $G\left(z\right)=\frac{\exp\left(z\right)}{\exp\left(z\right)+1}$. I understand that $G(z)$ is the algebraic result of manipulating $z=\ln\left(\frac{p}{p-1}\right)$ to isolate $P$ as the respondent. The textbook explains the interpretation of the coefficients in terms of p, the estimated probability of the dichotmous event, as $\frac{∂p}{∂x_i}\approx g\left(x\right)β_i$ for small changes in $x_i$. I understand this derivative equation as $∂G\left(z\right)\approx ∂z=∂xβ_i$ per the definition of $z$ above, and $g\left(z\right)=G'\left(z\right)=\frac{dG\left(z\right)}{d\left(z\right)}$, which is due to its natural log property. Are those equations accurate? Obviously, the change in $p$ given a change in $x_i$ won't be the same for all x . Therefore, in terms of $p$, the magnitude of the beta coefficient's effect on $p$ isn't constistent across x. The textbook and other sources seemed to leave it at that.

Now, in OTHER sources, I have found that most people will keep $g(•)$ on the left side of the equation, and interpret the betas as changes in the log odds instead. In this case, $\frac{∂y}{∂x}=β_i$, (where $y=\ln\left(\frac{p}{1-p}\right)$) for all x. Then, $exp(β)$ is the percent change in odds.

Is there a method/interpretation that is used more often academically and professionally? It seems that the log odds interpretation is more effective , seeing as it is consistent across all x.

Thanks

Tanner
  • 31
  • All you said are correct. Most used term/explanation is Odds Ratio = exp($\beta$). But based on log odds = $X\beta$, you need to derive the explanation required by different objectives of the studies. For example, if the objective is predict P(y=1|x), then you can get it from log odds. – user158565 Jul 14 '19 at 22:10
  • @user158565 so you are saying that you can derive the probability change from the change in log odds? – Tanner Jul 14 '19 at 22:23
  • Based on two different z, you can derive two probabilities from ln(p/(1-p))=z, then you can get the difference. Only based of the difference between two zs, you can not derive the difference of probabilities, because p is not linear function of z. – user158565 Jul 14 '19 at 22:29
  • @user158565 great. But the logodds term itself can be treated as linear, and with the exp(β) conversion to odds, beta can be interpreted as pct change in odds, correct? – Tanner Jul 14 '19 at 22:36
  • Yes, the log odds is linear function of $\beta$. We call exp($\beta$) odds ratio when x increase by 1 unit, because it is (odds|x+1)/(odds|(x)). Call it pct change may be inappropriate, because pct change means ((odds|x+1)-(odds|x))/(odds|x) * 100. – user158565 Jul 14 '19 at 22:45
  • @user158565 ok I think I understand. I said that it was pct change because ln(A) - ln(B)=ln(A/B)=β approximates pct change. Then, exp(β)=A/B≈1+ (A-B)/B. Does that make sense? – Tanner Jul 15 '19 at 00:37
  • It seems we have different explanation on pct change. Should be exp(β)=A/B=1+ (A-B)/B – user158565 Jul 15 '19 at 00:56
  • @user158565 Yeah that is what I said at the end. It is the ratio of the new value to the old – Tanner Jul 15 '19 at 00:57

0 Answers0